practice problem 1
The following four statements about circular orbits are equivalent. Derive any one of them from first principles.
- Negative kinetic energy equals half the potential energy (−K = ½U).
- Potential energy equals twice the total energy (U = 2E).
- Total energy equals negative kinetic energy (E = −K).
- Twice the kinetic energy plus the potential energy equals zero (2K + U = 0).
This is a key relationship for a larger problem in orbital mechanics known as the virial theorem
Circular orbits arise whenever the gravitational force on a satellite equals the centripetal force needed to move it with uniform circular motion.
Substitute this expression into the formula for kinetic energy.
Note how similar this new formula is to the gravitational potential energy formula.
|K = +
|Ug = −
The kinetic energy of a satellite in a circular orbit is half its gravitational energy and is positive instead of negative. When U and K are combined, their total is half the gravitational potential energy.
The gravitational field of a planet or star is like a well. The kinetic energy of a satellite in orbit or a person on the surface sets the limit as to how high they can "climb out of the pit". A satellite in a circular orbit is halfway out of the pit (or halfway in, for you pessimists).
practice problem 2
Determine the minimum energy required to place a large (five metric ton) telecommunications satellite in a geostationary orbit.
practice problem 3
practice problem 4